See this post for a somewhat detailed timeline.

But basically:

• They did not know for 16 hours that it was an atomic bomb, because the US did not announce it until then. Previously they just thought it was some kind of regular attack (the US had been mass-bombing Japanese cities for months by that point).

• The Japanese high command met and agreed to not act upon the announcement by the US until they confirmed it was true. So they sent a team of scientists to Hiroshima to see if it was real or just propaganda.

• It took a day for the Japanese scientists to get there and to inspect the city. By the end of August 8th (local time), they sent word back to Tokyo that it was indeed an atomic bomb.

• Overnight, the Soviet Union declared war on Japan and began their Manchurian invasion.

• The next morning the Japanese high command met to discuss the Soviet invasion and the atomic bomb. During this meeting they got notice of the attack on Nagasaki. (The plane that was involved in that attack had taken off around 1am Tokyo time.)

So there really was not very much time between the two — less than 3 days if you take into account the fact that they didn't even know it was meant to be an atomic bomb until most of the way through the first day, and even less if you take into account the fact that they (reasonably) wanted to confirm it was an atomic bomb before acting on the claim. Even less if you are asking in a way that might imply they could have done something that would have prevented the second attack (because the mission started in the middle of the night). I just bring this up because a lot of people justify the second bombing on the basis of, "well, they didn't surrender after the first one," but I don't think that amount of time is a reasonable amount of time if that was the goal (it wasn't). The original plan, for what it is worth, was to have 7 days between the two attacks, which is a more reasonable amount of time in my view. But weather conditions in Japan contracted the operation.

If I may add a follow-up question:

How much, if at all, did the fact that now 1 plane with 1 bomb could do what previously had required hundreds of planes and thousands of tons of bombs to do affect the thinking of the Japanese leaders?

My understanding is that Japan was running out of planes, fuel, and pilots. They couldn't send up planes to intercept every single USA plane that flew into Japanese airspace - at least not for long. But now, every single plane could destroy a city, so you pretty much have to send up an interceptor or be willing to lose a city.

Was this (or similar) thinking a factor at all? Or perhaps the military die hards were not convinced that this sort of thing was an issue? Or perhaps the Soviet invasion mooted it?